@Olivier5,
I agree. Godel may not have been "at odds with his own ideas". (The committee nature of "self" comes to mind here) I read somewhere that his belief was "intuitive" and emergent from "paradox" rather than subject to logical proof, irrespective of his own ontological attempts at such (finished by others, and published and criticized after his death). Also, I note that Godel used the term "properties" within his proof which indicates that he was assuming a paradigm of "naive realism" at the time as a setting for his usage of the term "existence". The interesting point, (to me at least
) is whether logic
per se shoots itself in the foot via the incompleteness theorem since the ontological
evocation (see above) of that appears to suggest that set membership properties are arbitrary,